Predictions for electron induced proton knockout from the $p_{1/2}$ and
$p_{3/2}$ shells in $^{16}$O are presented using various approximations for the
relativistic nucleonic current. Results for the differential cross section,
transverse-longitudinal response ($R_{TL}$) and left-right asymmetry $A_{TL}$
are compared at $|Q^2|=0.8$ (GeV/c)$^2$ corresponding to TJNAF experiment
89-003. We show that there are important dynamical and kinematical relativistic
effects which can be tested by experiment.Comment: 10 pages, including 2 figures. Removed preliminary experimental data
  from the figure

A. Picklesimer

B. D. Serot

C. Giusti

C. Mahaux

E. D. Cooper

E. Hummel

E. Moya de Guerra

G. H. Rawitscher

G. M. Spaltro

H. Müther

I. Bobeldijk

J. A. Caballero

J. E. Amaro

J. E. Ducret

J. J. Kelly

J. M. Udias

J. M. Udías

J. P. McDermott

L. Chinitz

L. Lapikas

M. Leuschner

M. M. Sharma

S. Boffi

S. Gilad

S. Jeschonnek

S. Ulrych

T. de Forest

T. W. Donnelly

V. R. Pandharipande

V. Van der Sluys

Y. Jin

Z. Y. Ma

English

arXiv

Crossref

Quasielastic Scattering from Relativistic Bound Nucleons: Transverse-Longitudinal Response

Predictions for electron induced proton knockout from p1/2 and p3/2 shells in 16O are presented using various approximations for the relativistic nucleonic current. Results for differential cross section, transverse-longitudinal response ( RTL), and left-right asymmetry ATL are compared at |Q2|=0.8(GeV/c)2. We show that there are important dynamical and kinematical relativistic effects which can be tested by experiment.Dirección General de Investigación Científica y Técnica PB/96-0604DGES PR156/9

Udías, J. M.

Caballero Carretero, Juan Antonio

Moya de Guerra, E.

Amaro Soriano, José Enrique

Donnelly, T. W.

idUS. Depósito de Investigación Universidad de Sevilla

VOLUME 83, NUMBER 26 P H Y S I C A L R E V I E W L E T T E R S 27 DECEMBER 1999Quasielastic Scattering from Relativistic Bound Nucleons: Transverse-Longitudinal ResponseJ. M. Udias,1,2 J. A. Caballero,1,3 E. Moya de Guerra,1 J. E. Amaro,4 and T. W. Donnelly51Instituto de Estructura de la Materia, CSIC Serrano 123, E-28006 Madrid, Spain2Departamento de Física Atómica, Molecular y Nuclear, Universidad Complutense de Madrid, E-28040 Madrid, Spain3Departamento de Física Atómica, Molecular y Nuclear, Universidad de Sevilla, E-41080 Sevilla, Spain4Departamento de Física Moderna, Universidad de Granada, E-18071 Granada, Spain5Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology,Cambridge, Massachusetts 02139(Received 26 April 1999)Predictions for electron induced proton knockout from p12 and p32 shells in 16O are presented usingvarious approximations for the relativistic nucleonic current. Results for differential cross section, trans-verse-longitudinal response (RTL), and left-right asymmetry ATL are compared at jQ2j  0.8GeVc2.We show that there are important dynamical and kinematical relativistic effects which can be tested byexperiment.PACS numbers: 25.30.Fj, 21.60.Cs, 24.10.– i, 25.30.RwThe exclusive e, e0p coincidence measurement allowsthe determination of the missing energy (Em) and missingmomentum ( pm, the momentum of the recoiling nucleus)in the reaction, and information on the energies,momentum distributions, and spectroscopic factors ofbound nucleons is provided [1,2]. Until recently low-Emdata were concentrated at pm # 300 MeVc. Now higherpm regions are being probed at small Em under quasielasticconditions [3], yielding new information of high momen-tum components of bound nucleons in a regime (Bjorkenx  1) where two-body currents can be safely neglected[4,5]. Most theoretical work on e, e0p has been basedon nonrelativistic approximations to the nucleon current,namely the standard distorted-wave impulse approximation(DWIA) [1]. DWIA data analyses [2] have met two majordifficulties: (a) The spectroscopic factors extracted fromlow-pm (pm , 300 MeVc) data are too small (for in-stance, Sa  0.5 for 3s12 and 2d52 orbits in 208Pb [2,6]),while theories on short-range correlations [7] predict atmost a 30% reduction of mean-field occupations (Sa .0.7) for levels just below the Fermi level. (b) DWIA cal-culations compatible with the low-pm data predict muchsmaller cross sections at high pm (pm . 300 MeVc) thanthose experimentally observed [6]. Short-range correla-tions are expected to increase the high-momentum compo-nents, but their effect is negligible [8] at the small missingenergies of these high-pm data [6], and long-range corre-lations have been introduced to explain the high-pm datawithin the nonrelativistic formalism [4,6].In recent years the relativistic mean-field approxima-tion has been sucessfully used for the analyses of bothlow-pm [9–13] and high-pm [14] data. In the relativisticdistorted-wave impulse approximation (RDWIA), the nu-cleon currentJmN v, q Zd p cF p 1 qĵmN v, qcB p (1)is calculated with relativistic cB (cF) wave function for0031-90079983(26)5451(4)$15.00the initial bound (final outgoing) nucleon. ĴmN is the rela-tivistic nucleon current operator of cc1 or cc2 forms as in[15]. For cB we use Dirac-Hartree solutions from relativis-tic Lagrangians with scalar and vector meson terms [16].For cF we compute a solution of the Dirac equation withscalar-vector (S-V) global optical potentials [17]. The onlyfitted parameter is the spectroscopic factor [10–13]. TheseRDWIA spectroscopic factors are larger than the DWIAones [11–13], and are valid both for low- and high-pmdata [14].Exploratory studies within the relativistic plane waveimpulse approximation (RPWIA), where the interactionin the final state (FSI) between the ejected nucleon andthe residual nucleus is neglected, showed [18–20] thatthe transverse-longitudinal response (RTL) is very sensi-tive to the negative-energy components of the relativisticbound nucleon wave function. In a recent work [19] wehave also shown that, for the j  l 6 12 spin-orbit part-ners of a given shell, this sensitivity is much larger for thej  l 2 12 than for the j  l 1 12 case. In this workwe observe for the first time this striking feature in a morerealistic description of the e, e0p reaction including rela-tivistic FSI.A certain degree of controversy surrounds the TL re-sponse measured in exclusive quasielastic electron scatter-ing from the least bound protons in 16O: in some cases [21]large deviations from standard DWIA calculations appear,while in others [9] the data are close to the calculations.New data on RTL for proton knockout from the 1p12 and1p32 orbits of 16O are available from Jefferson Laboratory(TJNAF) experiment 89-003 [3] at jQ2j  0.8 GeVc2.In this work we show that there are important kinematicaland dynamical relativistic effects for this case.We compute cB with the parameters of the set NL-SH[22]. For cF we solve the Dirac equation with theenergy-dependent, A-independent, potentials derived byClark et al. [17] for 16O. Using other relativistic potentialsavailable in the literature does not alter our conclusions.© 1999 The American Physical Society 5451VOLUME 83, NUMBER 26 P H Y S I C A L R E V I E W L E T T E R S 27 DECEMBER 1999The kinematical setup follows the TJNAF proposal [3]:beam energy 2445 MeV, j qj  1 GeVc, v  439 MeV,corresponding to quasielastic conditions (v  jQ2j2M),and proton kinetic energy of about 427 MeV.We divide the differences between this fully relativis-tic approach and the standard nonrelativistic one into twocategories: (i) Effects due to the relativistic 4-vector cur-rent operator, compared to the nonrelativistic one whichinvolves pM expansions. We call these effects kinemat-ical because they are independent of the dynamics intro-duced by the nuclear interaction. (ii) Effects due to thedifferences between relativistic and nonrelativistic nucleonwave functions, which depend on the 4-spinor structureand importantly on the potentials used in the respectiveDirac and Schrödinger equations. We call these effectsdynamical. We recall that in nonrelativistic approachesthe one-body current operator is expanded in a basis offree-nucleon plane waves, which amounts to a truncationof the nucleon propagator that ignores negative-energy so-lutions of the free Dirac equation.One may identify two types of relativistic dynamicaleffects: (a) Effects coming from the difference betweenthe upper components of cF , cB and the solutions ofthe Schrödinger equation. Assuming equivalent centraland spin-orbit potentials, this difference stems from thewell-known Darwin term. If the relativistic dynamics isknown, one can deduce the Darwin term and construct anequivalent bispinor wave function x to include its effectin the nonrelativistic nucleon current, thus removing thissource of difference between relativistic and nonrelativis-tic results [12,23]. This is done, for instance, in Ref. [24].The influence of this term on e, e0p observables has beendemonstrated in several works [11,12,25,26]. It appears tobe the main dynamical relativistic effect in the cross sec-tion in the low-pm region [11,12], and is important forthe correct determination of the spectroscopic factor fromlow-pm data. Its omission reduces the spectroscopic factorby 15%–20%. We included it in all calculations presentedhere. (b) The other dynamical effect is due to the nega-tive-energy components of the relativistic wave functions.Starting from Schrödinger-like solutions x one may at bestconstruct properly normalized four-spinors of the formc 1pN∑x p,s ? pÊ 1 Mx p∏(2)to calculate the relativistic nucleon current. This spinorlacks the dynamical enhancement due to the S-V potentialsof the lower component of the Dirac solution. This dynam-ical enhancement is present in the relativistic cB, cF so-lutions and influences e, e0p observables in the high-pmregions [12,18].Figure 1 shows the differential cross section, RTL re-sponse, and TL asymmetry (ATL) for p12 (top panels)and p32 (bottom panels). RTL and ATL are obtained fromthe difference of cross sections measured at fF  0± andfF  180± with the other variables v, Q2, Em, pm heldconstant, where fF is the azimuthal angle of the scattered5452proton (we follow the same convention as in [3]). Rela-tivistic calculations using the cc1 and cc2 current opera-tors are shown by solid and dotted lines, respectively. TheCoulomb gauge has been used throughout. The use of anoptical potential to account for inelastic reaction channelsbreaks Gauge invariance in DWIA. We estimated the un-certainty associated with the choice of gauge by comparingwith results in the Landau gauge. For the fully relativis-tic results at the kinematics of this work, the differencein going from Coulomb to Landau Gauge is very small,much smaller than the difference between cc1 and cc2 re-sults shown in the figures. We use a spectroscopic factorSa  0.7 as obtained for 208Pb [11], which in RDWIAalso matches the low-pm data [9,27] for the shells dis-cussed here; the spectroscopic factor simply scales downthe curves for differential cross sections and RTL whileleaving ATL unchanged.The role of the negative-energy components can be seenin Fig. 1. The short-dashed lines show the results obtainedwith the cc1 current operator when the negative-energycomponents are projected out, i.e., the nucleon current iscalculated asJmprojv, q Zd p c1F  p 1 qĴmv, qc 1B  p ,(3)where c1B (c1F ) is the positive-energy component ofcB (cF), i.e., c1B  p  L1 pcB p, L1 p M 1 p2M, with pm  pp2 1 M2, p (similarlyfor c1F ). The difference between the solid (cc1 full)and short dashed lines (cc1 projected) is due to thedynamical enhancement of the lower components whichis contained in the current of Eq. (1), but not in Eq. (3).This effect is more visible than that introduced by thetheoretical uncertainty due to the choice of cc1 (solidline) or cc2 (dotted line) operators. It is important torealize that the positive-energy projectors inserted inEq. (3) depend on the integration variable p. One mayattempt to neglect this dependence by using projectionoperators corresponding to asymptotic values of themomenta, i.e., projectors acting on cF and cB, res-pectively, with PmF  EF , pF, PmF 2 Qm the asymptoticfour-momentum of the outgoing and bound nucleon,respectively, with Qm  EF 2p pF 2 q2 1 M2, q.We refer to this approach as asymptotic projection (Jas).The corresponding results are shown by long dashed linesin Fig. 1. They are obtained with the cc1 operator andare very similar for cc2.All the differential cross sections for jpmj ,300 MeVc are similar, but for jpmj . 300 MeVcthey show a substantial dependence on the negative-en-ergy components for either the p12 or p32 shells. Thecross sections obtained with Jproj are more symmetricalaround pm  0 than the RDWIA results. Therefore, theeffect of removing the negative-energy components showsup more in RTL and ATL (see middle and right-handpanels of Fig. 1). The dependence on the dynamicalVOLUME 83, NUMBER 26 P H Y S I C A L R E V I E W L E T T E R S 27 DECEMBER 1999FIG. 1. Cross sections (left), RTL (middle), and ATL (right) for proton knockout from 16O for the 1p12 and 1p32 orbits versusmissing momentum pm. Solid (dotted) line shows the fully relativistic calculation using the Coulomb gauge and the current operatorcc1 (cc2). Also shown are the results after projecting the bound and scattered proton wave functions over positive-energy states(Jproj, short-dashed line) and using the asymptotic momenta (Jas, long-dashed line). For the p32 shell a small contribution from thenearby 521 and 121 states parametrized from the low-pm data [27] is included. Magnified plots of RTL for pm . 250 MeVcwhere accurate experimental data are being taken [3], are shown.enhancement of the lower components is stronger for thep12 RTL response than for p32, a feature that was firstseen in RPWIA [18] and that persists in this more realisticcalculation including FSI. Particularly interesting is theoscillatory structure of the fully relativistic result for ATL.This characteristic is preserved in the Jproj results, butnot in the Jas ones. The asymptotic projection severelymodifies ATL for both orbitals. We notice that the ATLcalculated with Jas are very similar to the ones obtainedin [24], particularly to the calculation called EMA(noSV)in said reference. At low momentum this approach isclose to the fully relativistic and to the Jproj results, butbeyond pm  200 MeVc it gives different results. Theoscillating trend of the ATL calculated in RDWIA is con-firmed by the preliminary data [3] and agrees qualitativelywith previous calculations by Van Orden [3]. As it wasthe case for RTL, the negative energy components clearlyaffect ATL for the j  l 2 12 partner, sizably drivingthe asymmetry towards negative values, even at low pm.Other relativistic effects can be seen in Fig. 2, wherewe compare RDWIA results on ATL (left panels) andRTL (right panels) to nonrelativistic approaches. Wehave used the cc2 current operator and nonrelativisticscattered wave functions obtained from Dirac-equivalentSchrödinger equations [12]. Thus the nonrelativistic wavefunctions correspond to the upper components of therelativistic ones, containing the Darwin term. For thenonrelativistic bound wave functions, we used the ones in[28] where new approximations to the on-shell relativisticone-body current operator were developed to accountbetter for relativistic kinematic effects in nonrelativisticcalculations. In particular, the charge density containsa spin-orbit correction r̂so that affects RTL [28,29]. InFig. 2 we show by long dashed lines the results obtainedwith the “relativized current” and by dotted lines theresults obtained when r̂so is neglected. r̂so has a largeeffect on RTL and ATL. Using the DWEEPY [30] code wehave obtained for ATL similar results to the dotted linesin Fig. 2.The short dashed lines in Fig. 2 are results obtainedwith the relativistic current and 4-spinors constructed as inEq. (2) from the nonrelativistic bound and scattered wavefunctions. In this way, all the elements of the relativisticcalculation but the dynamical enhancement of the lowercomponents are fully taken into account. This is why theseresults (short-dashed lines) for ATL and RTL are closer tothe fully relativistic results (solid lines). Contrary to theeffect of the dynamical enhancement of the lower com-ponents, the effects of relativistic kinematics are of thesame order in both shells. The data on ATL and RTL are astrong indication of the presence and crucial role played bydynamical effects of relativity affecting the lower compo-nents, in electron-nucleus observables. Strong sensitivityof the TL response to relativistic corrections was earlierfound for de, e0p [31].In conclusion, we have identified two types of relativis-tic effects on RTL and ATL. One is of kinematical origin,and has a large contribution from the spin-orbit correctionto the charge density, and the other is of dynamical origin.The latter is due to the enhancement of the lower compo-nents and is stronger for the p12 than for the p32 orbital.This is in addition to the dynamical effect on the upper5453VOLUME 83, NUMBER 26 P H Y S I C A L R E V I E W L E T T E R S 27 DECEMBER 1999FIG. 2. RTL (right panels) and ATL asymmetries (left panels)for proton knockout from 16O for the 1p12 (top panels) and1p32 (bottom panels) orbits. Results shown correspond to afully relativistic calculation using the Coulomb gauge and thecurrent operator cc2 (solid line), a calculation performed byprojecting the bound and scattered proton wave functions overpositive-energy states (short-dashed line) and two nonrelativisticcalculations with (long-dashed) and without (dotted) the spin-orbit correction term in the charge density operator (see text fordetails).component due to the Darwin term which is present in allthe results given here, and that mostly affects the determi-nation of spectroscopic factors [12]. It is encouraging thatthe preliminary data [3] agree so well with the predictionsof the fully relativistic calculations, and one anticipatesbeing able to make even more stringent tests when a finergrid of high-precision data involving other nuclei becomesavailable in the range 200 # pm # 400 MeVc.We thank the TJNAF Hall A collaboration, especiallyW. Bertozzi, J. Gao, S. Gilad, and J. J. Kelly for theirwillingness to discuss their data prior to publication, andfor many useful comments. This work was partially sup-ported under Contracts No. 940183 (NATO CollaborativeResearch Grant), No. DE-FC01-94ER40818 (cooperativeagreement with the U.S. Department of Energy DOE),No. PB/95-0123, No. PB/95-0533-A, No. PB/95-1204(DG-ICYT, Spain), No. PB/96-0604 (DGES, Spain),No. PR156/97 (Complutense University, Spain), and bythe Junta de Andalucía (Spain).[1] S. Boffi, C. Giusti, and F. D. Pacati, Phys. Rep. 226, 1(1993); S. Boffi, C. Giusti, F. Pacati, and M. Radici,5454Electromagnetic Response of Atomic Nuclei (ClarendomPress, Oxford, 1996); J. J. Kelly, Adv. Nucl. Phys. 23, 77(1996).[2] L. Lapikas, Nucl. Phys. A553, 297c (1993); E. Quint, Ph.D.thesis, NIKHEF, 1988.[3] T. Saha et al., TJNAF Proposal No. 89-003, 1989; J. Gao,Ph.D. thesis, MIT, 1999.[4] J. Ryckebush, D. Debruyne, W. N. Nespen, and S. Janssen,nucl-th/9904011; V. Van der Sluys, J. Ryckebusch, andM. Waroquier, Phys. Rev. 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Quasielastic scattering from relativistic bound nucleons: Transverse-longitudinal response

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.265.1994

Quaiselastic scattering from relativistic bound nucleons:
  Transverse-Longitudinal response

Quaiselastic scattering from relativistic bound nucleons: Transverse-Longitudinal response

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idUS. Depósito de Investigación Universidad de Sevilla